4. components of a life table

‫ לוחות תמותה שלמים של ישראל‬,‫למ"ס‬
This is a revised edition of Publication 1576, which presented life tables for 2008-2012. In the
previous edition, errors were found in the life tables due to biases in the estimates of
mortality rates at the oldest ages (over age 90). The new tables are based on a new method
of estimating probabilities of death after age 90, and extend the estimates from age 100 to
age 110+ (see Section 3, “Methods of Computation”).
The tables contain information on probabilities of death and life expectancy, including
standard deviation and confidence intervals. Data are presented by population group, sex,
and age.
The Central Bureau of Statistics produces two series of life tables on a regular basis:
abridged life tables,1 and complete life tables. The abridged life tables (by five-year age
groups) are produced for every calendar year, and the complete life tables (for single years
of age) are produced for periods of five calendar years (average). Data in the complete life
tables may differ from those in the abridged tables, especially in older age groups, owing to
differences in the methods of calculation (see Section 3: “Methods of Computation”, below).
The life expectancy at birth in 2008-2012 of the total population was 83.3 years for females
and 79.6 years for males. For Jews and Others, life expectancy was 83.7 years for females
and 80.1 years for males. In addition, the life expectancy of female Jews was 83.7 years, and
that of male Jews was 80.4 years. For Arabs, the life expectancy of females and males was
80.5 years and 76.3 years, respectively.
Based on the age-specific mortality rates in 2008-2012, more than half of the females born in
these years are expected to live more than 85 years, and more than half of the males born in
the same period are expected to live more than 82 years. In addition, 31.4% of the females
and 22.2% of the males born in 2008-2012 are expected to live at least 90 years.
Furthermore, based on these rates, females who reach the age of 65 can expect to live an
additional 21 years on the average, whereas those who have reached age 80 can expect to
live another 9.6 years on the average. Males who reach age 65 can expect to live 18.9 more
years on the average, and those who reach the age of 80 can expect to live another 8.8
years on the average.
Israeli males are ranked among the group of countries with the highest life expectancy.
According to the OECD,2 which presents data for the year 2011, Israeli males are in the fourth
place along with males of Sweden, and their life expectancy is 79.9 years; the countries with
higher life expectancy rates are Italy (80.1), Switzerland (80.5), and Iceland (80.7).
Israeli females rank lower among the OECD countries – in the thirteenth place – along with
Norway and Luxembourg, and their life expectancy is 83.6 years – 2.3 years less than that of
the leading country, Japan (85.9 years), and 2.1 years less than that of France (85.7), which
ranks second place.
See CBS. Statistical Abstract of Israel No. 63, 2013, Chapter 3: “Vital Statistics”, Tables 3.25–3.26.
Jerusalem: author, pp. 214–217.
OECD Health data 2013.
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‫ לוחות תמותה שלמים של ישראל‬,‫למ"ס‬
Types of Life Tables
There are two kinds of life tables: period tables and cohort tables.
The life tables presented in this publication are complete period life tables for single years of
age from birth (age 0) until age 110+.
Period life tables are meant to describe patterns of mortality for a specific period. A period
life table reflects the mortality of a hypothetical cohort born in a given year, assuming that
this generation will experience at each age the mortality rates existing for the same age
group during that year. For example, the life table for 1990 assumes that survivors of the
generation born in 1990 will be exposed at every age from 0 to 100+ to the same mortality
conditions that prevailed at the same ages from 0 to age 100+ in 1990. Thus, the calculation
is a sort of projection, assuming that mortality rates will remain constant.
Cohort (generational) life tables show mortality rates in a particular birth cohort until all
individuals in that cohort die. For example, the annual probabilities of deaths for persons
born in 1900 can be tracked until 2000, and their mortality rates can be obtained at every
age, from birth to age 100+. Based on these data, a life table can be compiled for the entire
cohort, assuming that all of them died by 2000. In order to produce a cohort life table,
mortality and immigration data have to be collected over a long period of time. This follow-up
is practical only among "closed" populations with no migration, which is far from the case in
Israel. Moreover, the value of a cohort table is mainly historical, because it reflects mortality
rates of individuals born long ago, who lived under different conditions from those prevailing
at the time the table was prepared.
Confidence Intervals
Mortality rates in Israel, as in all countries, are subject to stochastic variation (statistical
errors) and to a variety of non-stochastic errors, such as those that arise from mistaken
reports of year of birth or of age at death. Due to both kinds of errors, calculated mortality
rates may differ from the “true” mortality rate, which would have been obtained if it were
possible to overcome these errors. Stochastic variations are more significant when the
number of deaths is smaller, for example among small population groups or among a single
age group or over a short period of time.
This publication presents both standard deviation and confidence intervals for the probability
of death and for life expectancy. The confidence intervals are symmetrical; they reflect only
stochastic variation, and are based on the assumption that the number of deaths at each age
follows a binomial distribution.1
Chiang C.L. )1984(. Statistical Inference Regarding Life Table Functions. In C. L. Chiang, The Life
Table and its Applications (pp. 153-167). Malabar, FL: Robert E. Krieger Publishers.
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‫ לוחות תמותה שלמים של ישראל‬,‫למ"ס‬
A confidence interval of 95% represents a range in which the true value of the parameter will
be found in 95% of the cases. Whenever there is an overlap between the confidence
intervals of two probabilities for expected years of life between different ages or groups, the
differences are not statistically significant (at a confidence level of 95%).
The confidence interval of the probability of death (qx) is dependent on the number of deaths
in the reference group. Accordingly, there are differences in the relative width of the
confidence interval at different ages. At younger ages, in which there are fewer deaths, the
confidence interval is wider than at older ages, where there are more deaths. Similarly, the
relative width of the confidence interval differs among different population groups. Because
there are fewer deaths in the Arab population than in the Jewish population, the relative
width of the confidence intervals is greater among the Arabs.
The confidence interval of life expectancy is a function of the confidence interval of the
probability of death, and is therefore narrower for the Jewish population than for the Arab
population. For example, among Jewish females the confidence interval for life expectancy at
birth is (±) 0.1 years, compared with (±) 0.2 years for Arab females.
Confidence intervals for life expectancy and for probabilities of death were calculated using
the methods developed by Chiang,1 where the significance level α=0.05 corresponds to a
standardized normal distribution value of z=1.96. The confidence interval was calculated for
the estimated probability of death, which was obtained from the smoothed model (see
Section C – “Smoothing Techniques”, below).
Standard Deviation of the probability of death: S q x 
qˆ x2 (1  qˆ x )
Confidence interval: CI  2 * 1.96 * S qx
Standard Deviation of life expectancy:
Dx - Absolute number of deaths at age x.
Se x 
Tx - The total number of person-years lived by cohort survivors after reaching age x.
l x -The number of survivors at exact age x out of 100,000 infants born.
Chiang C.L. )1984(. Statistical Inference Regarding Life Table Functions. In C. L. Chiang, The Life
Table and its Applications (pp. 153-167). Malabar, FL: Robert E. Krieger Publishers.
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‫ לוחות תמותה שלמים של ישראל‬,‫למ"ס‬
Smoothing Techniques
Stochastic variation is not the only source of "error" in life table functions. Therefore, in order
to overcome irregularities from all sources, it is customary to use a “smoothing” technique of
some kind.
An “abridged” life table, which is based on mortality rates among broad age groups and not
on single years of age, is less exposed to stochastic variations and other errors.The
problems are more serious when calculating a “complete” life table based on single years of
age. Complete life tables in Israel for 1986-1990 until 1995-1999 were computed using the
MORTPAK1 software package, which was provided by the United Nations. The software
allows for calculation of complete life tables by estimating a Heligman-Pollard (H-P) mortality
model,2 by the least-squares method. Since 2000, it was found that this program does not
produce reasonable results for Israeli data. The fit between the model and the empirical data
is not statistically significant, and it was found that the H-P model raises life expectancy at
birth for all population groups (at least by 0.2 years and sometimes by more than one year)
as compared to the abridged life table. Moreover, it was found that the curve of the model
crosses the boundaries of the confidence interval for empirical probabilities of death (q x).
Furthermore, although the parameters of the H-P model can be estimated, the statistical
tools (standard deviation and significance) of the parameter estimates cannot be calculated.
Therefore, the overall statistical significance of the model is not known. Finally, this
smoothing procedure does not take into account the distinct features of the Israeli data: at
certain ages, the smoothing procedure greatly reduces the probability of death (for example,
the ages of compulsory military service) and at other ages (particularly at older ages), it
increases the probability.
For these reasons, a new method of smoothing was developed by means of a two-stage
polynomial function,3 and is used as the basis for the complete life tables since 1996-2000.
The model is based on the Local Maximum Likelihood method,4 as well as on a technique for
estimating change points.5
This method has four advantages:
a. The differences between the smoothed values of life expectancy and the original data
are not statistically significant.
b. Statistical parameters of the model, such as variance, confidence intervals, and
statistical significance can be estimated.
c. The model provides a good basis for smoothing qx (the specific probability of death at
a certain age) while considering the distinct features of the Israeli data.
d. The method is easy and convenient to use.
MORTPAK: for Windows Version 4.0. The United Nations Software Package for Demographic
Heligman L., & Pollard J. H. (1980). The Age Pattern of Mortality. Journal of the Institute of
Actuaries, 107, 49–75.
Vexler, A., Flaks, N., & Paltiel, A. (2005). A Method for Smoothing Mortality Functions Using a
Segmented Regression Model: An Application to Israeli Data. Working Paper Series No. 15.
Jerusalem: CBS (Hebrew only).
Fan, J., Farmen, M., & Gijbels I. (1998). Local Maximum Likelihood Estimation and Inference”.
Journal of the Royal Statistical Society, Series B, 60, 591-608.
Koul, H. L., Lianfen, Q., & Surgailis D. (2003). Asymptotics of M-Estimators in Two-Phase Linear
Regression Models. Stochastic Processes and their Applications, 103, 123-154.
‫ לוחות תמותה שלמים של ישראל‬,‫למ"ס‬
In the new method, life expectancy is calculated in four stages:
a. Calculation of the qx values based on mortality rates (mx) by singles years of age for
each population group and each sex, averaged for the five-year period (2008–2012).
b. The hypothesis that there is a change point in the model is tested. If the hypothesis is
not rejected one should proceed to the next stage.
c. The qx values are smoothed by estimating one or two models of the q x function,
depending on whether or not a change point was found, one for the younger ages (up
to the change point) and one for the older ages (after the change point).
d. All the parameters of the life table based on the model qx estimates are calculated.
Estimating Mortality Rates for Persons Aged 90 and Over
In recent years, it was found that the estimates of mortality rates at older ages obtained by
the above method were unreasonably low, both in terms of their rate of increase by age and
in comparison with mortality rates at younger ages. In Israel, random deviations as well as
errors in age reporting and possible errors in population estimates for persons age 90 and
over are higher than at younger ages. Therefore at these ages estimation based on a model
is a better way of providing stable and consistent estimates of mortality rates. In order to
estimate mortality rates at age 90-100, Kannisto’s logistic model1 was used. The model was
estimated using Maximum Likelihood methods, using a SAS Macro developed by Dr. Klára
Hulíková Tesárková.2 The Macro is based on the SAS NLIN procedure.
The formula for the model is:
m x   x 0.5 
 * e  *( x 0.5)
1   * e  *( x  0.5)
µx – the hazard rate (the likelihood of instantaneous death) at age x
α – represents the mortality rate age age 0
β – represents the (logistic) rate of increase in mortality from one age to the next.
The Macro estimates the α and β parameters on the basis of empirical mortality rates. The
model was estimated for all population groups and sexes, based on empirical mortality rates
from age 65 to 89; and probabilities of death in the life tables were calculated based on the
mortality rates provided by the model for ages 90 until 110+.
Kannisto V. (1994). The Development of Oldest-old Mortality 1950-1990: Evidence from 28
Developed Countries. Odense, Denmark: Odense University Press.
Hulíková Tesárková, K. (2012). Selected Methods of Mortality Analysis Focused on Adults and the
Oldest Age groups. Unpublished doctoral disservation, Department of Demography and
Geodemography, Charles University, Prague.
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‫ לוחות תמותה שלמים של ישראל‬,‫למ"ס‬
The life table is based on sex- and age-specific mortality rates, and consists of the following
Dx – Absolute number of deaths at age x.
mx – Average mortality rate at age x, i.e., the number of people who died at age x divided by
the average population at the same age. For example: the mx values for computing the
life table for 2008–2012 are based on average mortality rates for these years.
qx – The probability of death between age x and age x+1. The column presents the
proportion of people who died between age x and age x+1 of those living at age x. The
qx values are derived from mx values as follows:
qx 
1  1 mx
lx – The number of survivors at exact age x out of 100,000 infants born
(radix of the table – l0 = 100,000).
The lx values are based on the qx values, which allow for the calculation of the number
of survivors since age x-1.
lx = lx-1 (1- qx-1 )
Lx – The number of person-years of the cohort that reached exact age x, between this age
and age x+1.
Lx = (lx + lx+1)/2
L0 – The number of person-years lived by the cohort between birth and its first birthday.
L110+ – The number of person-years lived by the cohort from age 110 until the last one has
L0 and L110+ are calculated differently for two reasons:
L0 is affected by the non-linear distribution of deaths in the first year of life.
L110+ requires an estimate of the number of years that will be lived until the last member
of the cohort has died. Thus:
L0=0.3 l0+0.7 l1
L110+=1000 (l110/ m110+)
Tx – The total number of person-years lived by cohort survivors after reaching age x;
Tx is the sum of Lx for all ages after x.
ex – The life expectancy at age x. This is the average number of years a person may expect
to live after age x, assuming that he survived to age x, and that mortality rates are
ex 
The complete life tables presented here show the lx, qx and ex functions for single ages,
from birth to age 110+.
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